Since it's a bit silly to pay someone to do that for a humble chicken coop, and since learning how to do it sounds fun: have any of you developed plans for your own small structure and gotten them approved by your city? I'm in Los Angeles County.
Maybe in theory, you can design it yourself, in practice it may be painful and frustrating (or it may be easy as pie but "Los Angeles" makes me a bit doubtful).
To put it another way, you can design and build your own bicycle but their might be undesirable consequences of a lack of expertise/experience.
Disclaimer, I have a background in construction and architecture and once worked in a building department and before that a planning department. Good luck.